Internat. J. Math. & Math. Sci.
Vol. 22, No. 3 (1999) 617–628
S 0161-17129922617-8
© Electronic Publishing House
ON DECOMPOSABLE PSEUDOFREE GROUPS
DIRK SCEVENELS
(Received 16 October 1997)
Abstract. An Abelian group is pseudofree of rank if it belongs to the extended genus
of Z , i.e., its localization at every prime p is isomorphic to Zp . A pseudofree group can be
studied through a sequence of rational matrices, the so-called sequential representation.
Here, we use these sequential representations to study the relation between the product
of extended genera of free Abelian groups and the extended genus of their direct sum. In
particular, using sequential representations, we give a new proof of a result by Baer, stating
that two direct sum decompositions into rank one groups of a completely decomposable
pseudofree Abelian group are necessarily equivalent. On the other hand, sequential representations can also be used to exhibit examples of pseudofree groups having nonequivalent direct sum decompositions into indecomposable groups. However, since this cannot
occur when using the notion of near-isomorphism rather than isomorphism, we conclude
our work by giving a characterization of near-isomorphism for pseudofree groups in terms
of their sequential representations.
Keywords and phrases. Localization, extended genus, torsion-free Abelian group of finite
rank, sequential representation, direct sum decomposition, near-isomorphism.
1991 Mathematics Subject Classification. 20K15, 20K25, 20F18.
1. Introduction. The Mislin genus Ᏻ(N) [11] of a finitely generated nilpotent group
N is the set of isomorphism classes of finitely generated nilpotent groups M such that
the localization of M and N at every prime are isomorphic, that is Mp Np for all
primes p. If N is a finitely generated Abelian group, then N has a trivial Mislin genus.
This observation led Casacuberta and Hilton [2] to introduce the notion of extended
genus. They define the extended genus ᏱᏳ(N) of a finitely generated nilpotent group
N as the set of isomorphism classes of (not necessarily finitely generated) nilpotent
groups M such that Mp Np for all primes p. This notion admits interesting examples
as, for example, in [6], Hilton showed that there are uncountably many nonisomorphic
Abelian groups in the extended genus of Z.
Following the definition of extended genus, the notion of a pseudofree group was
introduced in [2, 6]. An Abelian group A is called pseudofree of rank —the name is
due to Ries [12]—if A belongs to the extended genus of Z . Clearly, a pseudofree group
of rank is torsion-free of rank . However, the fact that A is pseudofree imposes
strong restrictions on A. This in turn enables us to use methods which do not seem
to be applicable to the broad class of torsion-free Abelian groups of finite rank. In
particular, we obtain the so-called “sequential representation” of a pseudofree group
of rank , which consists of a sequence of invertible ( × )-matrices (one for each
prime) with entries in Q.
For a finitely generated nilpotent group N and for k ≥ 2, the function
618
DIRK SCEVENELS
k
Φ : Ᏻ(N) → Ᏻ N k
(1.1)
sending the k-tuple (L1 , . . . , Lk ) to L1 × · · · × Lk has been studied in [3, 7, 8]. If N has a
finite commutator subgroup, then Φ is surjective, but not necessarily injective. More
generally, for finitely generated nilpotent groups N1 , . . . , Nk (k ≥ 2), one can consider
an obvious function (cf. [7]),
Ψ : Ᏻ N1 × · · · × Ᏻ Nk → Ᏻ N1 × · · · × Nk .
(1.2)
If all the groups N1 , . . . , Nk belong to a certain subclass of the class of finitely generated nilpotent groups with finite commutator subgroup, then this function is again
surjective, but not necessarily injective (cf. [13]).
In Section 3, we study functions analogous to (1.1) and (1.2), now defined on the
extended genus. It then makes sense to consider these functions when the groups
involved are finitely generated and Abelian. As pointed out in [2], in this case, we may,
without loss of generality, restrict our attention to the functions
k
→ ᏱᏳ Zk
Φ̃ : ᏱᏳ Z
(1.3)
Ψ̃ : ᏱᏳ Z1 × · · · × ᏱᏳ Zk → ᏱᏳ Z1 +···+k ,
(1.4)
and
where k ≥ 2 and , 1 , . . . , k are positive integers. Of course, neither of these functions
is surjective since there exist pseudofree Abelian groups that are not decomposable.
If = 1, then we show that the function Φ̃ is injective up to a reordering of the factors
(see Corollary 3.2). In fact, this is a consequence of a well-known result due to Baer,
stating that two direct sum decompositions into rank one groups of a completely decomposable Abelian torsion-free group of finite rank are necessarily equivalent (cf. [4,
Thm. 86.1], [5, Thm. 117]). Using sequential representations, we give in Theorem 3.1
a new proof of Baer’s result for completely decomposable pseudofree groups of finite
rank. A proof for the case k = 2 can also be found in [9]. On the other hand, it is
well known that torsion-free Abelian groups of finite rank may have nonequivalent
direct sum decompositions into indecomposable groups (cf. [4]). This implies that, if
= 2, then Φ̃ is far from being injective. Indeed, in Example 2.3, we use sequential
representations to exhibit a pseudofree Abelian group A of rank 4 with nonequivalent
direct sum decompositions into indecomposable groups, i.e., A A1 ⊕ A2 B1 ⊕ B2 ,
where A1 A2 and B1 B2 are indecomposable pseudofree groups of rank 2, but
A1 B1 . As to the function Ψ̃ , in Example 2.4, we use sequential representations to
give a pseudofree Abelian group G of rank 3 such that G A⊕B C ⊕D, where A C
are pseudofree groups of rank 1 and B and D are nonisomorphic pseudofree groups
of rank 2. In the course of Section 3, we thus provide an answer to various questions
raised by Militello in [10].
Two torsion-free Abelian groups A and B of finite rank are called nearly isomorphic if, for every positive integer n, there is a subgroup An of B, of finite index
prime to n such that An A. Clearly, isomorphism implies near-isomorphism. The
above mentioned phenomena of nonequivalent direct sum decompositions, however,
ON DECOMPOSABLE PSEUDOFREE GROUPS
619
cannot appear if one uses the notion of near-isomorphism rather than isomorphism.
Indeed for torsion-free Abelian groups A, B, and C of finite rank, if A ⊕ C is nearly
isomorphic to B ⊕ C, then A is nearly isomorphic to B. Likewise, if Ak is nearly isomorphic to B k , then A is nearly isomorphic to B (cf. [1, Cor. 7.17]). In Section 4, we
therefore characterize nearly isomorphic pseudofree groups in terms of their sequential representations.
2. Pseudofree groups. For a finitely generated nilpotent group N, the extended
genus ᏱᏳ(N) [2] is the set of isomorphism classes of nilpotent groups M such that
Mp Np for all primes p. The Mislin genus Ᏻ(N) [11] is the subset of ᏱᏳ(N) containing
the isomorphism classes of finitely generated nilpotent groups. For short, we say that
a group M belongs to the extended (or Mislin) genus of N if the isomorphism class of
M does.
For a finitely generated Abelian group A, the Mislin genus Ᏻ(A) is easily seen to
be trivial. However, the situation for ᏱᏳ(A) is completely different. For example, the
extended genus of Z contains uncountably many isomorphism classes (cf. [6]). In fact,
in [2], Casacuberta and Hilton showed that in order to study the extended genus of any
finitely generated Abelian group, it suffices to consider ᏱᏳ(Z ). An Abelian group A is
called pseudofree of rank [12] if A belongs to ᏱᏳ(Z ). Note that if A is pseudofree of
rank , then A is torsion-free Abelian of rank . For convenience of the reader, we recall
from [2] the following basic facts about pseudofree groups. For a pseudofree group
A of rank , we can choose isomorphism f0 : A0 Q and fp : Ap Zp for all primes
p, which we write for short as {fp , p ≥ 0}. Since Ap is naturally embedded in A0 ,
the homomorphism f0 fp−1 : Zp → Q is actually a monomorphism. With respect to the
canonical basis {e1 , e2 , . . . , e } for Z , Zp , Q , this monomorphism f0 fp−1 is represented
by a matrix Mp ; explicitly,
f0 fp−1 ej = aij p ei ,
Mp = aij (p) ∈ GL Q .
(2.1)
i
We write M∗ for the sequence of matrices {Mpi }, where we enumerate the primes
as p1 , p2 , . . . , pi , . . . , and we call this sequence M∗ the sequential representation of A
associated with {fp , p ≥ 0}.
The pseudofree groups of rank one, the so-called groups of pseudo-integers, were
treated by Hilton in [6]. For a group of pseudo-integers, a sequential representation
simply consists of a sequence of rational numbers.
Example 2.1. Let A be a group of pseudo-integers. Then there exists a sequential
−m
representation M∗ of A of the form Mpi = (pi i ), where mi ≥ 0. In fact, A is iso −m
morphic to the subgroup of Q generated by the set pi i | all primes pi , and the
sequence (mi ) corresponds, in the classical sense, to a height sequence for a torsionfree Abelian group of rank one (cf. [1, 4, 5]).
A great deal of properties of pseudofree groups can be studied through their sequential representation. For future reference, we recall from [2] the following criterion for
two pseudofree groups to be isomorphic.
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DIRK SCEVENELS
Theorem 2.2. Let A, B be pseudofree groups of rank with respective sequential
representations M∗ and N∗ . Then A and B are isomorphic if and only if
(2.2)
∃Cp ∈ GL Zp , L ∈ GL Q : Np = LMp Cp for all p.
In the case of groups of pseudo-integers, this leads to the following (cf. [6]).
Example 2.3. Given groups of pseudo-integers A, B with respective sequential rep−m
−n
resentations M∗ and N∗ given by Mpi = (pi i ) and Npi = (pi i ). Then A is isomorphic
to B if and only if mi = ni for almost all i. This corresponds with the fact that the
isomorphism class of a torsion-free Abelian group of rank one is determined by its
type (cf. [1, Thm. 1.1] and [4, Thm. 85.1]).
Particular attention will be paid to completely decomposable pseudofree groups. Recall that a torsion-free Abelian group A of rank is called completely decomposable if
A=
Ai ,
(2.3)
i=1
where each Ai is torsion-free of rank one. Clearly, if A is a completely decomposable
pseudofree group of rank as in (2.3), then each Ai is a group of pseudo-integers.
Recall further (cf. [2, Thm. 4.3]) that a pseudofree group A of rank is completely
decomposable if and only if A admits a sequential representation D∗ , where each Dp
is a diagonal matrix.
3. Decomposable pseudofree groups. Let N be a finitely generated nilpotent group
with a finite commutator subgroup. From [7], we know that the Mislin genus Ᏻ(N) has
a natural Abelian group structure and that the function
k
Φ : Ᏻ(N) → Ᏻ N k
(fork ≥ 2)
(3.1)
given by
Φ L 1 , . . . , Lk = L1 × · · · × Lk
(3.2)
is actually a group homomorphism. In fact, Φ is a surjective homomorphism. This can
be seen as follows. Recall from [3] that the function
(3.3)
ρ : Ᏻ(N) → Ᏻ N k
given by
ρ(L) = L × N k−1
is a surjective homomorphism. Moreover, as explained in [7],
Φ L1 , . . . , Lk = L1 × · · · × Lk L1 + · · · + Lk × N k−1 = ρ L1 + · · · + Lk ,
(3.4)
(3.5)
where + denotes the operation in the Abelian group Ᏻ(N). Hence, since ρ is surjective,
we infer that Φ is surjective. In other words, every group in Ᏻ(N k ) is decomposable.
On the other hand, it was pointed out in [7] that Φ is far from being injective. Indeed,
in [7, Thm. 3.2], it is proved that, for Mi , Li ∈ Ᏻ(N) (i = 1, . . . , k), if M1 + · · · + Mk =
L1 + · · · + Lk in Ᏻ(N), then M1 × · · · × Mk L1 × · · · × Lk . Moreover, in [8], the kernel of
ON DECOMPOSABLE PSEUDOFREE GROUPS
621
Φ is precisely described in the case where N belongs to a certain subclass of the class
of finitely generated nilpotent groups with finite commutator subgroup.
We now confine our attention to the analogue of the function Φ, defined on the
extended genus rather than the Mislin genus. As pointed out in [2], in order to study
this function in the case when the groups involved are finitely generated Abelian, it
suffices to consider the case of Z . Thus, for ≥ 1 and k ≥ 2, we consider the function
k
→ ᏱᏳ Zk
Φ̃ : ᏱᏳ Z
(3.6)
Φ̃ L1 , . . . , Lk = L1 × · · · × Lk .
(3.7)
given by
Consider the case where = 1. It was pointed out in [2] that Φ̃ : ᏱᏳ(Z)k → ᏱᏳ(Zk ) is
not surjective. Indeed, the image of Φ̃ consists precisely of those pseudofree Abelian
groups of rank k that are completely decomposable. As to the injectivity of Φ̃, a classical result by Baer (cf. [4, Prop. 86.1], [5, Thm. 117]) states that two direct sum decompositions into rank one groups of a completely decomposable torsion-free Abelian
group of finite rank are necessarily equivalent. This means that if A is a completely
decomposable Abelian torsion-free group of rank k and
A
k
Ai i=1
k
Bi ,
(3.8)
i=1
where Ai , Bi (i = 1, . . . , k) are torsion-free of rank one, then there exists a permutation
σ ∈ Σk such that Ai Bσ (i) for all i. Using sequential representations, we here give a
new proof of this fact for pseudofree groups.
Theorem 3.1. Let A be a completely decomposable pseudofree group of rank k. Suppose that
A
k
Ai i=1
k
Bi ,
(3.9)
i=1
where Ai , Bi (i = 1, . . . , k) are groups of pseudo-integers. Then there exists a permutation
σ ∈ Σk such that Ai Bσ (i) for all i ∈ {1, . . . , k}.
Proof. We may assume that Ai 1/p mi (p) , all p and Bi 1/p ni (p) , all p for
i = 1, . . . , k. Then clearly A has a sequential representation M∗ given by

1
 p m1 (p)


 0


Mp = 
..


.



0
0
1
p m2 (p)
0
···
0
···
0
..
..
.
.
1
p mk (p)







,





(3.10)
622
DIRK SCEVENELS
but A also has a sequential representation N∗ given by
 1

0
···
0 
 p n1 (p)




1
 0

·
·
·
0


p n2 (p)



.
Np =
.
.
..
 .

.
 .

.
.





1 
0
0
p nk (p)
(3.11)
Theorem 2.2 then implies that there exist L ∈ GLk (Q) and Cp ∈ GLk (Zp ) such that
Np = LMp Cp for all primes p. Equivalently, L−1 Np = Mp Cp for all p. If we put L−1 =
(ij ) and Cp = (cij (p)), then this is equivalent to
cji (p)
ji
= m (p)
p ni (p)
p j
(3.12)
for all i, j ∈ {1, . . . , k} and for all primes p. Since det Cp is a unit in Zp and since the
entries of Cp belong to Zp , we infer that there exists a permutation σ ∈ Σk such that
cσ (i)i (p) is a unit in Zp for all i. Moreover, since, for almost all p, the nonzero ji are
units in Zp , we conclude from (3.12) that
mσ (i) (p) = ni (p)
(3.13)
for almost all p. This implies that
Ai Bσ (i)
fori ∈ {1, . . . , k}.
(3.14)
Corollary 3.2. Let Φ̃ be as in (3.6) with = 1. Then
Φ̃ L1 , . . . , Lk = Φ̃ M1 , . . . , Mk
(3.15)
if and only if (L1 , . . . , Lk ) and (M1 , . . . , Mk ) belong to the same orbit under the obvious
action of Σk on (ᏱᏳ(Z))k .
However, as the following example shows (cf. [1, Ex. 2.11]), the situation is completely different if we consider Φ̃ defined on the extended genus of Z2 .
Example 3.3. Let P1 ∪ P2 ∪ {5} be a partition of the set of all primes, where P1 and
P2 are infinite. Consider the pseudofree group A of rank 4 given by the sequential
representation

 
1

 
1
1
0
0
0
0
0
1

0
0
0 
 
5
p

 
 1

 
1
 
1



 


0
0
0
0
0
0
 0



1
0
0 

p2



 
5



,
(3.16)
Mp = 
,


 0
1
1
0
1
0  
 0



0
0
1


0
0

 
 
5
p1


 
1  

1
0
0
0
0
0
0
1
0
0
0
p2
5
for respectively p = p1 ∈ P1 , p = p2 ∈ P2 , and p = 5. Consider also the pseudofree
623
ON DECOMPOSABLE PSEUDOFREE GROUPS
group B of rank 4 given by the sequential representation

1
p
 1

 0

Np = 

 0


0
0
0
1
0
0
0
 
1
0 

 
 0
0 
 
,
 
0
0
 
 

0
1
1
p1
0
0
1
p2
0
0
 
1
 
 
 
0
0 
 
,
 
0  
0
 
 
1  

0
p2
0
0
0
1
0
1
5
2
5
0
0
0
1
0
0

0


0



1

5

2
(3.17)
5
for respectively p = p1 ∈ P1 , p = p2 ∈ P2 , and p = 5. Then clearly A A1 ⊕ A2 , where
A1 A2 is an indecomposable pseudofree group of rank 2 with sequential representation
 
1
0 
,

0
1

1

Xp =  p1
0
 
0
 1
 
1 ,
0
p2

1
5

1
(3.18)
5
for respectively p = p1 ∈ P1 , p = p2 ∈ P2 , and p = 5. Analogously, B B1 ⊕ B2 , where
B1 B2 is an indecomposable pseudofree group of rank 2 with sequential representation
 
1
0 
,

0
1

1

Yp =  p1
0
0
 
 1
 
1 ,
0
p2

1
5

2
(3.19)
5
for respectively p = p1 ∈ P1 , p = p2 ∈ P2 , and p = 5. Observe that A B. Indeed,
setting

1

0
L=
0

0
0
2
0
−5
3
0
−1
0

0

1

0

−2
(3.20)
1
−4
−2
10
3
0
−1
0

1

−2

−1

4,
(3.21)
and Cp = L−1 for all p ≠ 5, while

1

0
C5 = 
0

0
it is easily verified that Np = LMp Cp for all primes p. On the other hand, note that
A1 B1 . Indeed, suppose that there exist L ∈ GL2 (Q) and Cp ∈ GL2 (Zp ) such that
LXp Cp = Yp for all primes p. If we set
x
L=
y
z
,
t
(3.22)
624
DIRK SCEVENELS
then we infer that Cp−1 = Yp−1 LXp is equal to

x


y
p1
 
  x
,

t
p2 y
p1 z


1
z
x − y
2


p2  , 
 5
t
y
2

x +z y +t
−

5
10 


y +t
2
(3.23)
for respectively p = p1 ∈ P1 , p = p2 ∈ P2 , and p = 5. This implies that p1 | y and p2 | z
for all p1 ∈ P1 and for all p2 ∈ P2 . Hence, y = 0 = z. This means that

x
C5−1 = 

0

t
x
−
5 10 
.

t
Moreover, since, for all p different from 5, we
x
Cp−1 =
0
(3.24)
2
have that
0
,
t
(3.25)
we infer that x = 5i and t = 5j for some integers i, j. However, this is in contradiction
with C5−1 ∈ GL2 (Z5 ).
Another function of interest considered in [7] is the following. Let N1 , . . . , Nk be
finitely generated nilpotent groups with a finite commutator subgroup and consider
Ψ : Ᏻ N1 × · · · × Ᏻ Nk → Ᏻ N1 × · · · × Nk
(3.26)
Ψ L 1 , . . . , Lk = L1 × · · · × L k .
(3.27)
given by
In [7], it is shown that Ψ is a group homomorphism and the authors asked whether Ψ
is always surjective. In [13], this homomorphism Ψ is studied for a certain subclass of
the class of finitely generated nilpotent groups with finite commutator subgroup. In
this case, Ψ is indeed surjective and, moreover, the exact conditions for Ψ (L1 , . . . , Lk ) =
Ψ (M1 , . . . , Mk ) (i.e., for L1 × · · · × Lk M1 × · · · × Mk ) to hold are described.
Again, we consider the analogue function defined on the extended genus and we are
particularly interested in
Ψ̃ : ᏱᏳ Z1 × · · · × ᏱᏳ Zk → ᏱᏳ Z1 +···+k .
(3.28)
Of course, this function Ψ̃ is not surjective. Indeed, not every pseudofree group of
finite rank is decomposable. Furthermore, Ψ̃ is far from being injective in general.
Indeed, it is a well-known fact that torsion-free Abelian groups of finite rank may
have nonequivalent direct sum decompositions (cf. [4, Thm. 90.4]), In fact, we can
already exhibit a pseudofree group of rank 3 that has two nonequivalent direct sum
decompositions (cf. [1, Ex. 2.10]).
Example 3.4. Let P1 ∪ P2 ∪ {5} be a partition of the set of all primes, where P1 and
P2 are infinite. Consider the pseudofree group G of rank 3 given by the sequential
625
ON DECOMPOSABLE PSEUDOFREE GROUPS
representation
 1
 p1


Mp =  0


0
0
1
p1
0
 
0 1
 
 0
,
0
 
 
0
1
0
1
0
 
1
 
 

0 
0
,
 
1  

0
p2
0
0
1
0
0



1

5

1
(3.29)
5
for respectively p = p1 ∈ P1 , p = p2 ∈ P2 , and p = 5. Consider also the pseudofree
group H of rank 3 given by the sequential representation

 8
 
 
5
1 0 3
1 0
0
0

 p1 p1
 
 


 
 
6
 3
 
 0 1
0


2
0 1



,
Np = 
(3.30)
,

0 
5



1  
 p1 p1
 


1
0 0
0 0
0
0
1
p2
5
for respectively p = p1 ∈ P1 , p = p2 ∈ P2 , and p = 5. Then it is easily seen that G A ⊕ B, where A is pseudofree of rank 1 with sequential representation (1/p1 ), (1), (1)
for respectively p = p1 ∈ P1 , p = p2 ∈ P2 , and p = 5, and B is pseudofree of rank 2
with sequential representation
 

 

1
1
0
1
1



0
 

5
 

(3.31)
,
 p1
1 ,
1
0
0
1
0
p2
5
for respectively p = p1 ∈ P1 , p = p2 ∈ P2 , and p = 5. Analogously, it can be verified that H C ⊕ D, where C is pseudofree of rank 1 with sequential representation (1/p1 ), (1), (1) for respectively p = p1 ∈ P1 , p = p2 ∈ P2 , and p = 5, and D is
pseudofree of rank 2 with sequential representation
 

 

3
1
1
0
1
0 
 
p
5
 

(3.32)
,
 1

1 ,
1
0
0
1
0
p2
5
for respectively p = p1 ∈ P1 , p = p2 ∈ P2 , and p = 5. Finally, observe that G H and
A C while B D.
4. Near-isomorphism of pseudofree groups. Motivated by the fact that the above
phenomena of nonequivalent direct sum decompositions of a pseudofree group cannot appear if we use the notion of near-isomorphism rather than the notion of isomorphism (cf. [1, Cor. 7.17]), we give in this section a characterization of near-isomorphism
for pseudofree groups in terms of their sequential representations.
Recall (see, e.g., [1]) that two torsion-free Abelian groups A and B of finite rank are
called nearly isomorphic (notation A n B) if, for every positive integer n, there is a
subgroup An of B, of finite index prime to n such that An A. Note that A and B
are then necessarily of the same rank. Moreover, two nearly isomorphic torsion-free
Abelian groups obviously belong to the same extended genus.
Theorem 4.1. Let A and B be pseudofree groups of rank k with respectively sequential representations M∗ and N∗ , induced by {fp , p ≥ 0} and {fp , p ≥ 0}. Then A n B
626
DIRK SCEVENELS
if and only if, for all positive integers n, there exists Φn ∈ GLk (Q) such that Np−1 Φn Mp
has entries in Zp for all primes p, and Np−1 Φn Mp ∈ GLk (Zp ) for almost all primes p,
including all the prime divisors of n.
Proof. Suppose that A n B. For all positive integers n, there exists an embedding
jn : A → B of finite index such that [B : jn (A)] = kn is prime to n. Then (jn )0 : A0 → Bo
is an isomorphism and we may choose (fn )0 : A0 → Qk , (fn )0 : B0 → Qk such that the
following diagram commutes.
(fn )0
A0
(jn )0
)
(fn
0
B0
/ Qk
(4.1)
id
/ Qk ,
i.e., (fn )0 = (fn )0 ◦ (jn )0 . This means that we change the sequential representations
M∗ and N∗ to respectively M̃∗ , Ñ∗ given by
Ñp = Sn Np ,
M̃p = Sn Mp ,
(4.2)
where Sn , Sn ∈ GLk (Q). Since now the conditions of [2, Thm. 2.6] are satisfied, we
infer that
−1
M̃p
(4.3)
Ñp
has entries in Zp for all primes p. This is equivalent to
Np−1 Φn Mp
(4.4)
−1
having entries in Zp for all p, where Φn = Sn Sn . Moreover, since [B : jn (A)] = kn
is prime to n, we know that (jn )p : Ap → Bp is an isomorphism for all primes p not
dividing kn . If p does not divide kn , then we have the following diagram:
Zkp o
(fn )p
Ap

/ A0
(jn )p
Zkp o
)
(fn
p

B
/ Qk
(jn )0
/ B0
p
(fn )0
)
(fn
0
(4.5)
id
/ Qk .
Hence, if p does not divide kn , then there exists an isomorphism γn (p) : Zkp → Zkp
−1
such that (fn )0 ◦(fn )−1
p ◦γn (p) = (fn )0 ◦(fn )p . This means that the matrix of γn (p),
given by
−1
S n Np
Sn Mp = Np−1 Φn Mp ,
(4.6)
belongs to GLk (Zp ) if p kn .
Conversely, let the isomorphisms φn : Qk → Qk correspond to the given matrices
Φn . Then the composition
Zkp
fp−1
/ Ap 
/ A0
f0
/ Qk
φn
/ Qk
(4.7)
corresponds to a sequential representation for A, given by Φn Mp , and induced by
ON DECOMPOSABLE PSEUDOFREE GROUPS
627
homomorphisms (gn )0 := φn ◦ f0 and (gn )p := fp . By [2, Thm. 2.6], we find a homomorphism jn : A → B such that (gn )0 = f0 ◦ (jn )0 . Moreover, due to the hypothesis
on Np−1 Φn Mp , we deduce the existence of a monomorphism γn (p) : Zkp → Zkp , which
is an isomorphism for almost all primes p, including all prime divisors of n. Hence,
it induces a monomorphism jn (p) : Ap → Bp (which is an isomorphism for almost all
primes, including the prime divisors of n ) such that the following diagram commutes.
Zkp o
γn (p)
Zkp o
fp
Ap

/ Ao
(gn )0
jn (p)

B
fp
/ B0
p
f0
/ Qk
(4.8)
id
/ Qk .
Now set jn (0) = fo−1 ◦ (gn )0 . Then the diagram
Ap

jn (p)

B
p
/ A0
(4.9)
jn (0)
/ B0
commutes for all primes p. Hence, there exists a monomorphism jn : A → B such that
(jn )p = jn (p), which is an isomorphism for almost all p, including the prime divisors
of n. Finally, it is easily verified that jn is, in fact, an embedding of finite index prime
to n.
Example 4.2. With the notations as in Example 2.3, we show that A1 n B1 . Let n
be any positive integer and suppose that its prime divisors are p1 , . . . , pk . Choose an
integer m such that 2 + 5m is a prime number different from p1 , . . . , pk and set
1
0
.
(4.10)
Φn =
0 2 + 5m
Then it is easy to verify that Yp−1 Φn Xp has entries in Zp for all primes p and that it
belongs to GL2 (Zp ) for almost all primes, including p1 , . . . , pk .
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Scevenels: Centre de Recerca Matemàtica, Apartat 50, E-08193 Bellaterra, Spain